Constructive Wall-Crossing
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The Particle Zoo
219 8 The Particle Zoo 8.1 Introduction Around 1960 the situation in particle physics was very confusing. Elementary particlesa such as the photon, electron, muon and neutrino were known, but in addition many more particles were being discovered and almost any experiment added more to the list. The main property that these new particles had in common was that they were strongly interacting, meaning that they would interact strongly with protons and neutrons. In this they were different from photons, electrons, muons and neutrinos. A muon may actually traverse a nucleus without disturbing it, and a neutrino, being electrically neutral, may go through huge amounts of matter without any interaction. In other words, in some vague way these new particles seemed to belong to the same group of by Dr. Horst Wahl on 08/28/12. For personal use only. particles as the proton and neutron. In those days proton and neutron were mysterious as well, they seemed to be complicated compound states. At some point a classification scheme for all these particles including proton and neutron was introduced, and once that was done the situation clarified considerably. In that Facts And Mysteries In Elementary Particle Physics Downloaded from www.worldscientific.com era theoretical particle physics was dominated by Gell-Mann, who contributed enormously to that process of systematization and clarification. The result of this massive amount of experimental and theoretical work was the introduction of quarks, and the understanding that all those ‘new’ particles as well as the proton aWe call a particle elementary if we do not know of a further substructure. -
22 Scattering in Supersymmetric Matter Chern-Simons Theories at Large N
2 2 scattering in supersymmetric matter ! Chern-Simons theories at large N Karthik Inbasekar 10th Asian Winter School on Strings, Particles and Cosmology 09 Jan 2016 Scattering in CS matter theories In QFT, Crossing symmetry: analytic continuation of amplitudes. Particle-antiparticle scattering: obtained from particle-particle scattering by analytic continuation. Naive crossing symmetry leads to non-unitary S matrices in U(N) Chern-Simons matter theories.[ Jain, Mandlik, Minwalla, Takimi, Wadia, Yokoyama] Consistency with unitarity required Delta function term at forward scattering. Modified crossing symmetry rules. Conjecture: Singlet channel S matrices have the form sin(πλ) = 8πpscos(πλ)δ(θ)+ i S;naive(s; θ) S πλ T S;naive: naive analytic continuation of particle-particle scattering. T Scattering in U(N) CS matter theories at large N Particle: fund rep of U(N), Antiparticle: antifund rep of U(N). Fundamental Fundamental Symm(Ud ) Asymm(Ue ) ⊗ ! ⊕ Fundamental Antifundamental Adjoint(T ) Singlet(S) ⊗ ! ⊕ C2(R1)+C2(R2)−C2(Rm) Eigenvalues of Anyonic phase operator νm = 2κ 1 νAsym νSym νAdj O ;ν Sing O(λ) ∼ ∼ ∼ N ∼ symm, asymm and adjoint channels- non anyonic at large N. Scattering in the singlet channel is effectively anyonic at large N- naive crossing rules fail unitarity. Conjecture beyond large N: general form of 2 2 S matrices in any U(N) Chern-Simons matter theory ! sin(πνm) (s; θ) = 8πpscos(πνm)δ(θ)+ i (s; θ) S πνm T Universality and tests Delta function and modified crossing rules conjectured by Jain et al appear to be universal. Tests of the conjecture: Unitarity of the S matrix. -
Old Supersymmetry As New Mathematics
old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee Atiyah-Singer Index Theorem ~ 1963 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) Calabi-Yau ~ 1978 1977 ~ Supersymmetry Calibrated Geometry ~ 1982 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) (Harvey & Lawson) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 1994 ~ 4d Wall-Crossing (Seiberg & Witten) (Kontsevich) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) Wall-Crossing Conjecture ~ 2008 2008 ~ Konstevich-Soibelman Explained (conjecture by Kontsevich & Soibelman) (Gaiotto & Moore & Neitzke) 2011 ~ KS Wall-Crossing proved via Quatum Mechanics (Manschot , Pioline & Sen / Kim , Park, Wang & P.Y. / Sen) 2012 ~ S2 Partition Function as tt* (Jocker, Kumar, Lapan, Morrison & Romo /Gomis & Lee) quantum and geometry glued by superstring theory when can we perform path integrals exactly ? counting geometry with supersymmetric path integrals quantum and geometry glued by superstring theory Einstein this theory famously resisted quantization, however on the other hand, five superstring theories, with a consistent quantum gravity inside, live in 10 dimensional spacetime these superstring theories say, spacetime is composed of 4+6 dimensions with very small & tightly-curved (say, Calabi-Yau) 6D manifold sitting at each and every point of -
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory. -
Horizon Crossing Causes Baryogenesis, Magnetogenesis and Dark-Matter Acoustic Wave
Horizon crossing causes baryogenesis, magnetogenesis and dark-matter acoustic wave She-Sheng Xue∗ ICRANet, Piazzale della Repubblica, 10-65122, Pescara, Physics Department, Sapienza University of Rome, P.le A. Moro 5, 00185, Rome, Italy Sapcetime S produces massive particle-antiparticle pairs FF¯ that in turn annihilate to spacetime. Such back and forth gravitational process S, FF¯ is described by Boltzmann- type cosmic rate equation of pair-number conservation. This cosmic rate equation, Einstein equation, and the reheating equation of pairs decay to relativistic particles completely deter- mine the horizon H, cosmological energy density, massive pair and radiation energy densities in reheating epoch. Moreover, oscillating S, FF¯ process leads to the acoustic perturba- tions of massive particle-antiparticle symmetric and asymmetric densities. We derive wave equations for these perturbations and find frequencies of lowest lying modes. Comparing their wavelengths with horizon variation, we show their subhorion crossing at preheating, and superhorizon crossing at reheating. The superhorizon crossing of particle-antiparticle asymmetric perturbations accounts for the baryogenesis of net baryon numbers, whose elec- tric currents lead to magnetogenesis. The baryon number-to-entropy ratio, upper and lower limits of primeval magnetic fields are computed in accordance with observations. Given a pivot comoving wavelength, it is shown that these perturbations, as dark-matter acoustic waves, originate in pre-inflation and return back to the horizon after the recombination, pos- sibly leaving imprints on the matter power spectrum at large length scales. Due to the Jeans instability, tiny pair-density acoustic perturbations in superhorizon can be amplified to the order of unity. Thus their amplitudes at reentry horizon become non-linear and maintain approximately constant physical sizes, and have physical influences on the formation of large scale structure and galaxies. -
Quantum Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction .................................. -
On the Possibility of Generating a 4-Neutron Resonance with a {\Boldmath $ T= 3/2$} Isospin 3-Neutron Force
On the possibility of generating a 4-neutron resonance with a T = 3/2 isospin 3-neutron force E. Hiyama Nishina Center for Accelerator-Based Science, RIKEN, Wako, 351-0198, Japan R. Lazauskas IPHC, IN2P3-CNRS/Universite Louis Pasteur BP 28, F-67037 Strasbourg Cedex 2, France J. Carbonell Institut de Physique Nucl´eaire, Universit´eParis-Sud, IN2P3-CNRS, 91406 Orsay Cedex, France M. Kamimura Department of Physics, Kyushu University, Fukuoka 812-8581, Japan and Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan (Dated: September 26, 2018) We consider the theoretical possibility to generate a narrow resonance in the four neutron system as suggested by a recent experimental result. To that end, a phenomenological T = 3/2 three neutron force is introduced, in addition to a realistic NN interaction. We inquire what should be the strength of the 3n force in order to generate such a resonance. The reliability of the three-neutron force in the T = 3/2 channel is exmined, by analyzing its consistency with the low-lying T = 1 states of 4H, 4He and 4Li and the 3H+ n scattering. The ab initio solution of the 4n Schr¨odinger equation is obtained using the complex scaling method with boundary conditions appropiate to the four-body resonances. We find that in order to generate narrow 4n resonant states a remarkably attractive 3N force in the T = 3/2 channel is required. I. INTRODUCTION ergy axis (physical domain) it will have no significant impact on a physical process. The possibility of detecting a four-neutron (4n) structure of Following this line of reasoning, some calculations based any kind – bound or resonant state – has intrigued the nuclear on semi-realistic NN forces indicated null [13] or unlikely physics community for the last fifty years (see Refs. -
TO the POSSIBILITY of BOUND STATES BETWEEN TWO ELECTRONS Alexander A
WEPPP031 Proceedings of IPAC2012, New Orleans, Louisiana, USA TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS Alexander A. Mikhailichenko, Cornell University, LEPP, Ithaca, NY 14853, USA Abstract spins for reduction of minimal emittance restriction arisen from Eq. 1. In some sense it is an attempt to prepare the We analyze the possibility to compress dynamically the pure quantum mechanical state between just two polarized electron bunch so that the distance between electrons. What is important here is that the distance some electrons in the bunch comes close to the Compton between two electrons should be of the order of the wavelength, arranging a bound state, as the attraction by Compton wavelength. We attracted attention in [2,3] that the magnetic momentum-induced force at this distance attraction between two electrons determined by the dominates repulsion by the electrostatic force for the magnetic force of oppositely oriented magnetic moments. appropriately prepared orientation of the magnetic In this case the resulting spin is zero. Another possibility moments of the electron-electron pair. This electron pair considered below. behaves like a boson now, so the restriction for the In [4], it was suggested a radical explanation of minimal emittance of the beam becomes eliminated. structure of all elementary particles caused by magnetic Some properties of such degenerated electron gas attraction at the distances of the order of Compton represented also. wavelength. In [5], motion of charged particle in a field OVERVIEW of magnetic dipole was considered. In [4] and [5] the term in Hamiltonian responsible for the interaction Generation of beams of particles (electrons, positrons, between magnetic moments is omitted, however as it protons, muons) with minimal emittance is a challenging looks like problem in contemporary beam physics. -
Vacuum Energy
Vacuum Energy Mark D. Roberts, 117 Queen’s Road, Wimbledon, London SW19 8NS, Email:[email protected] http://cosmology.mth.uct.ac.za/ roberts ∼ February 1, 2008 Eprint: hep-th/0012062 Comments: A comprehensive review of Vacuum Energy, which is an extended version of a poster presented at L¨uderitz (2000). This is not a review of the cosmolog- ical constant per se, but rather vacuum energy in general, my approach to the cosmological constant is not standard. Lots of very small changes and several additions for the second and third versions: constructive feedback still welcome, but the next version will be sometime in coming due to my sporadiac internet access. First Version 153 pages, 368 references. Second Version 161 pages, 399 references. arXiv:hep-th/0012062v3 22 Jul 2001 Third Version 167 pages, 412 references. The 1999 PACS Physics and Astronomy Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 11.10.+x, 04.62.+v, 98.80.-k, 03.70.+k; The 2000 Mathematical Classification Scheme: http://www.ams.org/msc 81T20, 83E99, 81Q99, 83F05. 3 KEYPHRASES: Vacuum Energy, Inertial Mass, Principle of Equivalence. 1 Abstract There appears to be three, perhaps related, ways of approaching the nature of vacuum energy. The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant. -
Crossing Symmetry in the Planar Limit
Crossing Symmetry in the Planar Limit Sebastian Mizera [email protected] Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Crossing symmetry asserts that particles are indistinguishable from anti-particles traveling back in time. In quantum field theory, this statement translates to the long- standing conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this work we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta one can smoothly interpolate between pairs of crossing channels without encountering singularities or violating mass-shell conditions and momentum conservation. The analytic continuation can be realized using two types of moves. The first one makes use of an i" prescription for avoiding singularities near the physical kinematics and allows us to adjust the momenta of the external particles relative to one another within their lightcones. The second, more violent, step involves a rotation of subsets of particle momenta via their complexified lightcones from the future to the past and vice versa. We show that any singularity along such a deformation would have to correspond to two beams of particles scattering off each other. For planar Feynman diagrams, these kinds of singularities are absent because of the particular flow of energies through their propagators. -
Bound and Quasi Bound States in the Continuum
Bound and Quasi-Bound States in the Continuum One-dimensional models of square-integrable states embedded in continua, first offered by von Neumann and Wigner, can stabilize ordinary resonance levels and have gained importance in connection with solid state heterostructure semi conductor devices. General continuum bound state theories for many-particle systems do not exist, but models for continuum crossings of bound states suggest possible long-lived states, including some double charged anions. Nonlinear varia tional calculations can provide estimates of resonance energies and of the rapid onset of instability in various atomic and molecular ionization phenomena, typifying results found in mathematical "catastrophe theory". One-Dimensional Models :Many years ago von Neumann and Wignerl pointed out that local potentials could have bound (i.e. square-integrable) eigenstates with positive energies. These states are embedded in the continuum of scattering states with the same symmetry, but nevertheless would fail to "ionize". Thus it would be proper to regard these continuum bound states as infinitely sharp resonances. The example offered by von Neumann and Wigner (subsequently corrected by Simon2) was one-dimensional, and involved an oscillatory potential V(x) which behaved essentially as (sin x)/x for large lxl. Likewise the square-integ rable wavefunction l/;(x) for the positive energy bound state was oscillatory but, owing to diffractive interference induced by V(x), the amplitude of l/1 was driven toward zero as lxl increased, so that particle binding resulted. The exist ence of this elementary example naturally raises questions about whether similar continuum bound states could exist in physically more realistic cases.